Francisco Naranjo

Rybakov's theorem for vector measures in Fréchet spaces

Abstract For a real Frechet space X with dual X′ the following conditions are equivalent: 1. (a) X admits a continuous norm. 2. (b) Every convex and weakly compact subset of X is the closed convex hull of its exposed points. 3. (c) For every X-valued, countably additive measure ν there exists x′ in X′ such that ν is ¦x′ v¦- continuous .

Complex interpolation of spaces of integrable functions with respect to a vector measure

Let (Ω, Σ) be a measurable space andm: Σ →X be a vector measure with values in the complex Banach space X: We apply the Calderón interpolation methods to the family of spaces of scalarp-integrable functions with respect tom with 1≤p≤∞. Moreover we obtain a result about the relation between the complex interpolation spaces [X0,X1][θ] and [X0,X1][θ] for a Banach couple of interpolation (X0,X1) such thatX1 ⊂X0 with continuous inclusion.

AL- and AM-spaces of integrable scalar functions with respect to a Fréchet-valued measure

We study the question of determining conditions for the space of R-valued integrable functions with respect to a vector measure taking its values in a real Fréchet space to be an AL- or an AM-space.

Compactness of Multiplication Operators on Spaces of Integrable Functions with Respect to a Vector Measure

We study properties of compactness of multiplication operators between spaces of p-power integrable scalar functions with respect to a vector measure m.

Norming sets and integration with respect to vector measures

Abstract Let v be a countably additive measure defined on a measurable space (Ω, Σ) and taking values in a Banach space X. Let f : Ω → ℝ be a measurable function. In order to check the integrability (respectively, weak integrability) of f with respect to v it is sometimes enough to test on a norming set Λ ⊂ X*. In this paper we show that this is the case when A is a James boundary for B X * (respectively, Λ is weak*-thick). Some examples and applications are given as well.

Operators and the space of integrable scalar functions with respect to a Fréchet-valued measure

We consider the space L 1 (ν, X ) of all real functions that are integrable with respect to a measure v with values in a real Frechet space X . We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of l ∞ in the space of all linear continuous operators from a complete DF-space Y to a Frechet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure ν and the space X that imply that L 1 (ν, X ) has the Dunford-Pettis property. Applications of these results to Frechet AL-spaces and Kothe sequence spaces are also given.

Strictly positive linear functional and representation of Fréchet lattices with the Lebesgue property

Abstract For Frechet lattices with the Lebesgue property and weak order unit we prove that the following conditions are equivalent: 1. (1) there exists a strictly positive linear functional, 2. (2) the Frechet lattice admits a continuous norm. In this setting we obtain a representation theorem for Frechet lattices.